## Conservation of Energy First Principle of Thermodynamics

30 The first principle of thermodynamics relates the work done on a (closed) system and the heat transfer into the system to the change in energy of the system. We shall assume that the only energy transfers to the system are by mechanical work done on the system by surface traction and body forces, by heat transfer through the boundary. 6.4.1 Spatial Gradient of the Velocity 31 We define L as the spatial gradient of the velocity and in turn this gradient can be decomposed into a symmetric rate...

## External Work

19 External work W performed by the applied loads on an arbitrary system is defined as where b is the body force vector t is the applied surface traction vector and r is that portion of the boundary where t is applied, and u is the displacement. 20 For point loads and moments, the external work is 21 For linear elastic systems, (P KA) we have for point loads When this last equation is combined with Pf KAf we obtain When this last equation is combined with Pf KAf we obtain where K is the...

## Indicial Notation

32 Whereas the Engineering notation may be the simplest and most intuitive one, it often leads to long and repetitive equations. Alternatively, the tensor and the dyadic form will lead to shorter and more compact forms. 33 While working on general relativity, Einstein got tired of writing the summation symbol with its range of summation below and above (such as aijbi) and noted that most of the time the upper range (n) was equal to the dimension of space (3 for us, 4 for him), and that when the...

## Internal Strain Energy

Ii The strain energy density of an arbitrary material is defined as, Fig. 13.1 The complementary strain energy density is defined The strain energy itself is equal to 14 To obtain a general form of the internal strain energy, we first define a stress-strain relationship accounting for both initial strains and stresses where D is the constitutive matrix (Hooke's Law) e is the strain vector due to the displacements u e0 is the initial strain vector a0 is the initial stress vector and a is the...

## Stress Strain Relations

29 In orthogonal curvilinear coordinates, the physical components of a tensor at a point are merely the Cartesian components in a local coordinate system at the point with its axes tangent to the coordinate curves. Hence, For Plane strain problems, from Eq. 7.75 and zz Yrz YOz Trz Tgz 0. 31 Inverting, 32 For plane stress problems, from Eq. 7.78-a and Trz TOz & zz Yrz YOz 0 33 Inverting

## Stress Strain Relations in Generalized Elasticity Anisotropic

28 From Eq. 7.22 and 7.23 we obtain the stress-strain relation for homogeneous anisotropic material which is Hooke's law for small strain in linear elasticity. 29 We also observe that for symmetric cij we retrieve Clapeyron formula 30 In general the elastic moduli cij relating the cartesian components of stress and strain depend on the orientation of the coordinate system with respect to the body. If the form of elastic potential function W and the values cij are independent of the orientation,...

## Traction on an Arbitrary Plane Cauchys Stress Tensor

7 Let us now consider the problem of determining the traction acting on the surface of an oblique plane (characterized by its normal n) in terms of the known tractions normal to the three principal axis, 11, t2 and t3. This will be done through the so-called Cauchy's tetrahedron shown in Fig. 2.3. 8 The components of the unit vector n are the direction cosines of its direction n1 cos( Z AON) n2 cos(Z BON) n3 cos(Z CON) (2.3) The altitude ON, of length h is a leg of the three right triangles...

## Victor Saouma Ecublens Juin

One of the most fundamental question that a Material Scientist has to ask him herself is how a material behaves under stress, and when does it break. Ultimately, it its the answer to those two questions which would steer the development of new materials, and determine their survival in various environmental and physical conditions. The Material Scientist should then have a thorough understanding of the fundamentals of Mechanics on the qualitative level, and be able to perform numerical...

## Potential Energy Derivation

47 From section , if U0 is a potential function, we take its differential 49 We now define the variation of the strain energy density at a point1 Applying the principle of virtual work, Eq. 13.37, it can be shown that 1 Note that the variation of strain energy density is, Uo amp ij, and the variation of the strain energy itself is amp U fn U0dQ. Figure 13.4 Single DOF Example for Potential Energy 51 We have thus derived the principle of stationary value of the potential energy Of all...

## Force Traction and Stress Vectors

1 There are two kinds of forces in continuum mechanics body forces act on the elements of volume or mass inside the body, e.g. gravity, electromagnetic fields. dF pbdVol. surface forces are contact forces acting on the free body at its bounding surface. Those will be defined in terms of force per unit area. 2 The surface force per unit area acting on an element dS is called traction or more accurately stress vector. I tdS if txdS j tydS k tzdS 2.1 Most authors limit the term traction to an...

## Mathematica Assignment and Solution

Connect to Mathematica using the following procedure 5. On the newly opened shell, enter your password first, and then type setenv DISPLAY xxx 0.0 where xxx is the workstation name which should appear on a small label on the workstation itself. and then solve the following problems 1. The state of stress through a continuum is given with respect to the cartesian axes 0x1x2x3 by Determine the stress vector at point P 1,1, a 3 of the plane that is tangent to the cylindrical surface x2 X3 4 at P....

## Load Shear Moment Relations

38 Let us derive the basic relations between load, shear and moment. Considering an infinitesimal length dx of a beam subjected to a positive load5 w x , Fig. 12.6. The infinitesimal section must also be in equilibrium. 39 There are no axial forces, thus we only have two equations of equilibrium to satisfy XFy 0 and XMz 0. 40 Since dx is infinitesimally small, the small variation in load along it can be neglected, therefore we assume w x to be constant along dx. 41 To denote that a small change...

## Principle of Virtual Work and Complementary Virtual Work

33 The principles of Virtual Work and Complementary Virtual Work relate force systems which satisfy the requirements of equilibrium,, and deformation systems which satisfy the requirement of compatibility. 1. In any application the force system could either be the actual set of external loads dp or some virtual force system which happens to satisfy the condition of equilibrium p. This set of external forces will induce internal actual forces da or internal hypothetical forces Sa compatible with...

## Geometric Instability

33 The stability of a structure is determined not only by the number of reactions but also by their arrangement. 34 Geometric instability will occur if 1. All reactions are parallel and a non-parallel load is applied to the structure. 2. All reactions are concurrent, Fig. 12.4. Figure 12.4 Geometric Instability Caused by Concurrent Reactions Figure 12.4 Geometric Instability Caused by Concurrent Reactions 3. The number of reactions is smaller than the number of equations of equilibrium, that is...

## Transversely Isotropic Material

38 A material is transversely isotropic if there is a preferential direction normal to all but one of the three axes. If this axis is x3, then rotation about it will require that cos 6 sin 6 0 sin 6 cos 6 0 0 0 1 substituting Eq. 7.33 into Eq. 7.41, using the above transformation matrix, we obtain C1111 cos4 6 c1111 cos2 6 sin2 6 2cm2 4cm2 sin4 6 c2222 C1122 cos2 6 sin2 6 cm1 cos4 6 cm2 4 cos2 6 sin2 6 cm2 sin4 6 c2211 sin2 6 cos2 6 c2222 cos2 6 c1133 sin2 6 c2233 sin4 6 cm1 cos2 6 sin2 6 2cm2...

## Example Mohrs Circle in Plane Stress

An element in plane stress is subjected to stresses axx 15, ayy 5 and Txy 4. Using the Mohr's circle determine a the stresses acting on an element rotated through an angle 0 40 counterclockwise b the principal stresses and c the maximum shear stresses. Show all results on sketches of properly oriented elements. Solution 1. The center of the circle is located at 2 axx ayy 1 15 5 10. 2.63 2. The radius and the angle 2p are given by R y1 15 5 2 42 6.403 2.64-a 2 4 tan2 w 0.8 38.66 f3 19.33 2.64-b...

## Summary

60 Summary of Virtual work methods, Table 13.2. Displacement strains Forces Stresses KAD Kinematically Admissible Dispacements SAS Statically Admissible Stresses Table 13.2 Comparison of Virtual Work and Complementary Virtual Work 61 A summary of the various methods introduced in this chapter is shown in Fig. 13.7. 62 The duality between the two variational principles is highlighted by Fig. 13.8, where beginning with kinematically admissible displacements, the principle of virtual work provides...

## Boundary Conditions

4 In describing the boundary conditions B.C. , we must note that 1. Either we know the displacement but not the traction, or we know the traction and not the corresponding displacement. We can never know both a priori. 2. Not all boundary conditions specifications are acceptable. For example we can not apply tractions to the entire surface of the body. Unless those tractions are specially prescribed, they may not necessarily satisfy equilibrium. 5 Properly specified boundary conditions result...

## Principal Values and Directions of Symmetric Second Order Tensors

59 Since the two fundamental tensors in continuum mechanics are of the second order and symmetric stress and strain , we examine some important properties of these tensors. 60 For every symmetric tensor Tij defined at some point in space, there is associated with each direction specified by unit normal nj at that point, a vector given by the inner product If the direction is one for which vi is parallel to ni, the inner product may be expressed as and the direction ni is called principal...

## Cauchys Reciprocal Theorem

21 If we consider t1 as the traction vector on a plane with normal n1, and t2 the stress vector at the same point on a plane with normal n2, then t1 n1 and t2 n2a ti L 1JM and 2 L -2JH If we postmultiply the first equation by n2 and the second one by n1, by virtue of the symmetry of 0 we have Figure 2.4 Cauchy's Reciprocal Theorem Figure 2.4 Cauchy's Reciprocal Theorem 22 In the special case of two opposite faces, this reduces to 23 We should note that this theorem is analogous to Newton's...

## Size Effect Griffith Theory

13 In his quest for an explanation of the size effect, Griffith came across Inglis's paper, and his strike of genius was to assume that strength is reduced due to the presence of internal flaws. Griffith postulated that the theoretical strength can only be reached at the point of highest stress concentration, and accordingly the far-field applied stress will be much smaller. 14 Hence, assuming an elliptical imperfection, and from equation 11.2 a is the stress at the tip of the ellipse which is...

## Isotropic Material

40 An isotropic material is symmetric with respect to every plane and every axis, that is the elastic properties are identical in all directions. 41 To mathematically characterize an isotropic material, we require coordinate transformation with rotation about x2 and xi axes in addition to all previous coordinate transformations. This process will enforce symmetry about all planes and all axes. 42 The rotation about the x2 axis is obtained through cos 6 0 sin 6 0 1 0 sin 6 0 cos 6 we follow a...

## Example Stress Vector normal to the Tangent of a Cylinder

The stress tensor throughout a continuum is given with respect to Cartesian axes as Determine the stress vector or traction at the point P 2, VS of the plane that is tangent to the cylindrical surface x2 x2 4 at P, Fig. 3.9. Figure 3.9 Radial Stress vector in a Cylinder Figure 3.9 Radial Stress vector in a Cylinder At point P, the stress tensor is given by The unit normal to the surface at P is given from Thus the traction vector will be determined from 20 We can also define the gradient of a...

## Operations

Addition of two vectors a b is geometrically achieved by connecting the tail of the vector b with the head of a, Fig. 1.2. Analytically the sum vector will have components a b a2 b2 a3 b3 J. Scalar multiplication aa will scale the vector into a new one with components aai aa2 aa3 J. Vector Multiplications of a and b comes in three varieties Dot Product or scalar product is a scalar quantity which relates not only to the lengths of the vector, but also to the angle between them. a-b a UK b cos9...

## V

Finally, the displacement components can be obtained by integrating the above equations 10.2.2.3 Example Thick-Walled Cylinder 4i If we consider a circular cylinder with internal and external radii a and b respectively, subjected to internal and external pressures p and po respectively, Fig. 10.2, then the boundary conditions for the plane strain problem are Ttt Pi at r Oi Ttt -po at r b 42 These Boundary conditions can be easily shown to be satisfied by the following stress field These...

## Rotation of Axes

55 The rule for changing second order tensor components under rotation of axes goes as follow ajTjqvq From Eq. 1.39-a 1.71 But we also have Ui Tipvp again from Eq. 1.39-a in the barred system, equating these two expressions we obtain By extension, higher order tensors can be similarly transformed from one coordinate system to another. 56 If we consider the 2D case, From Eq. 1.38 2 sin 2aTxx sin 2aTyy 2 cos 2aTxy 2 sin 2aTxx sin 2aTyy 2 cos 2aTxy sin2 aTxx cos a cos aTyy 2 sin aTxy 0...

## J Contravariant Transformation

12 The vector representation in both systems must be the same V Vqbq Vkbk Vk bkbq v - Vk b bq 0 1.23 since the base vectors bq are linearly independent, the coefficients of bq must all be zero hence showing that the forward change from components vk to vq used the coefficients bqk of the backward change from base bq to the original bk. This is why these components are called contravariant. 13 Generalizing, a Contravariant Tensor of order one recognized by the use of the superscript transforms a...

## Rayleigh Ritz Method

55 Continuous systems have infinite number of degrees of freedom, those are the displacements at every point within the structure. Their behavior can be described by the Euler Equation, or the partial differential equation of equilibrium. However, only the simplest problems have an exact solution which satisfies equilibrium, and the boundary conditions . 56 An approximate method of solution is the Rayleigh-Ritz method which is based on the principle of virtual displacements. In this method we...

## Arch Elasticity

51 Fig. 2.10 illustrates the stresses acting on a differential element of a shell structure. The resulting forces in turn are shown in Fig. 2.11 and for simplification those acting per unit length of the middle surface are shown in Fig. 2.12. The net resultant forces are given by Figure 2.9 Mohr Circle for Stress in 3D Figure 2.10 Differential Shell Element, Stresses Figure 2.10 Differential Shell Element, Stresses Figure 2.11 Differential Shell Element, Forces Figure 2.11 Differential Shell...

## Spherical and Deviatoric Stress Tensors

36 If we let o denote the mean normal stress p 0 -p 3 011 022 033 3on tr a then the stress tensor can be written as the sum of two tensors Hydrostatic stress in which each normal stress is equal to p and the shear stresses are zero. The hydrostatic stress produces volume change without change in shape in an isotropic medium. Deviatoric Stress which causes the change in shape.

## Linear Momentum Principle Equation of Motion Momentum Principle

22 The momentum principle states that the time rate of change of the total momentum of a given set of particles equals the vector sum of all external forceps acting on the particles of the set, provided Newton's Third Law applies. The continuum form of this principle is a basic postulate of continuum mechanics. i tdS pbdV d f pvdV 6.22 Then we substitute ti Tijnj and apply the divergence theorm to obtain which is Cauchy's first equation of motion, or the linear momentum principle, or more...

## Continuum Mechanics

Saouma Exam I Closed notes , March 27, 1998 3 Hours There are 19 problems worth a total of 63 points. Select any problems you want as long as the total number of corresponding points is equal to or larger than 50. 1. 2 pts Write in matrix form the following 3rd order tensor Dijk in R2 space. i,j,k range from 1 to 2. 2. 2 pts Solve for Eijai in indicial notation. 3. 4 pts if the stress tensor at point P is given by determine the traction or stress vector t on the plane passing...

## Mohrs Circle for Plane Stress Conditions

41 The Mohr circle will provide a graphical mean to contain the transformed state of stress axx,ayy, xy at an arbitrary plane inclined by a in terms of the original one axx, ayy, axy . cos2 a 1 c s2a sin2 a -c s2 cos 2a cos2 a - sin2 a sin 2a 2 sin a cos a into Eq. 2.49 and after some algebraic manipulation we obtain 7 axx Vyy T. Jxx - yy cos2a aXy sin2a 2.57-a TXy axy cos 2a - ctxx - yy sin2a 2.57-b 43 Points axx,axy , axx, 0 , ayy, 0 and axx ayy 2, 0 are plotted in the stress representation...

## Navier Cauchy Equations

11 One such approach is to substitute the displacement-strain relation into Hooke's law resulting in stresses in terms of the gradient of the displacement , and the resulting equation into the equation of motion to obtain three second-order partial differential equations for the three displacement components known as Navier's Equation Figure 9.3 Fundamental Equations in Solid Mechanics

## Cartesian Coordinate System

17 If we consider two different sets of cartesian orthonormal coordinate systems ei, e2, e3 and e, e2, e3 , any vector v can be expressed in one system or the other is To determine the relationship between the two sets of components, we consider the dot product of v with one any of the base vectors 19 We can thus define the nine scalar values which arise from the dot products of base vectors as the direction cosines. Since we have an orthonormal system, those values are nothing else than the...

## Shear Moment and Deflection Diagrams for BEAMS

Adapted from 1 Simple Beam uniform Load 2 Simple Beam Unsymmetric Triangular Load 3 Simple Beam Symmetric Triangular Load 3 Simple Beam Symmetric Triangular Load 4 Simple Beam Uniform Load Partially Distributed 4 Simple Beam Uniform Load Partially Distributed 5 Simple Beam Concentrated Load at Center at x 2 when x lt 2 whenx lt 2 at x L 6 Simple Beam Concentrated Load at Any Point

## Strain Decomposition

72 In this section we first seek to express the relative displacement vector as the sum of the linear Lagrangian or Eulerian strain tensor and the linear Lagrangian or Eulerian rotation tensor. This is restricted to small strains. 73 For finite strains, the former additive decomposition is no longer valid, instead we shall consider the strain tensor as a product of a rotation tensor and a stretch tensor. 4.3.1 jLinear Strain and Rotation Tensors 74 Strain components are quantitative measures of...

## Euler Equation

1 The fundamental problem of the calculus of variation1 is to find a function u x such that 2 We define u x to be a function of x in the interval a, b , and F to be a known function such as the energy density . 3 We define the domain of a functional as the collection of admissible functions belonging to a class of functions in function space rather than a region in coordinate space as is the case for a function . 4 We seek the function u x which extremizes n. 5 Letting u to be a family of...

## Principle of Virtual Work

35 Derivation of the principle of virtual work starts with the assumption of that forces are in equilibrium and satisfaction of the static boundary conditions. The Equation of equilibrium Eq. 6.26 which is rewritten as where b representing the body force. In matrix form, this can be rewritten as Note that this equation can be generalized to 3D. 37 The surface r of the solid can be decomposed into two parts r and T where tractions and displacements are respectively specified. t t on rt Natural...

## Saint Venants Principle

Is This famous principle of Saint Venant was enunciated in 1855 and is of great importance in applied elasticity where it is often invoked to justify certain simplified solutions to complex problem. In elastostatics, if the boundary tractions on a part r of the boundary r are replaced by a statically equivalent traction distribution, the effects on the stress distribution in the body are negligible at points whose distance from r is large compared to the maximum distance between points of ri....

## Principal Strains Strain Invariants Mohr Circle

So Determination of the principal strains E 3 lt E 2 lt E i , strain invariants and the Mohr circle for strain parallel the one for stresses Sect. 2.4 and will not be repeated here. where the symbols IE, IIE and IIIE denote the following scalar expressions in the strain components Ie En E22 E33 En tr E 4.164 IIe E11E22 E22E33 E33E11 E223 E31 E222 4.165 2 EjEij EiiEjj 2Eij Eij 2IE 4.166 IIIe detE eijk epqr EipEjqEkr 4.168 87 In terms of the principal strains, those invariants can be simplified...

## Statics Equilibrium

8 Any structural element, or part of it, must satisfy equilibrium. 1 So far we have restricted ourselves to a continuum, in this chapter we will consider a structural element. Summation of forces and moments, in a static system must be equal to zero2. In a 3D cartesian coordinate system there are a total of 6 independent equations of equilibrium SFx SFy SF 0 SMx SMy SM2 0 11 In a 2D cartesian coordinate system there are a total of 3 independent equations of equilibrium 12 All the externally...

## Principle of Complementary Virtual Work

40 Derivation of the principle of complementary virtual work starts from the assumption of a kinematicaly admissible displacements and satisfaction of the essential boundary conditions. 41 Whereas we have previously used the vector notation for the principle of virtual work, we will now use the tensor notation for this derivation. 42 The kinematic condition strain-displacement 43 The essential boundary conditions are expressed as 44 The principle of virtual complementary work or more...

## Coordinate Transformation jGeneral Tensors

9 Let us consider two bases bj x x3 and bj xi, x2x3 , Fig. 1.5. Each unit vector in one basis must be a linear combination of the vectors of the other basis bj- apbp and bk b bg 1.20 summed on p and q respectively where apP subscript new, superscript old and bk are the coefficients for the forward and backward changes respectively from b to b respectively. Explicitly Figure 1.5 Coordinate Transformation Figure 1.5 Coordinate Transformation 10 The transformation must have the determinant of its...

## Introduction

1 In Eq. we showed that around a circular hole in an infinite plate under uniform traction, we do have a stress concentration factor of 3. 2 Following a similar approach though with curvilinear coordinates , it can be shown that if we have an elliptical hole, Fig. , we would have We observe that for a b, we recover the stress concentration factor of 3 of a circular hole, and that for a degenerated ellipse, i.e a crack there is an infinite stress. Alternatively, the stress can be expressed in...

## Lagrangian Stresses Piola Kirchoff Stress Tensors

96 In Sect. 2.2 the discussion of stress applied to the deformed configuration dA using spatial coordiantes x , that is the one where equilibrium must hold. The deformed configuration being the natural one in which to characterize stress. Hence we had note the use of T instead of a . Hence the Cauchy stress tensor was really defined in the Eulerian space. 97 However, there are certain advantages in referring all quantities back to the undeformed configuration Lagrangian of the body because...

## Strain Tensor

8 Following the simplified and restrictive introduction to strain, we now turn our attention to a rigorous presentation of this important deformation tensor. 9 The approach we will take in this section is as follows 1. Define Material fixed, Xj and Spatial moving, Xj coordinate systems. 2. Introduce the notion of a position and of a displacement vector, U, u, with respect to either coordinate system . 3. Introduce Lagrangian and Eulerian descriptions. 4. Introduce the notion of a material...

## Cartesian Coordinates Plane Strain

16 If the deformation of a cylindrical body is such that there is no axial components of the displacement and that the other components do not depend on the axial coordinate, then the body is said to be in a state of plane strain. If e3 is the direction corresponding to the cylindrical axis, then we have Ui Ui xi,x2 , U2 U2 xi,X2 , U3 0 and the strain components corresponding to those displacements are and the non-zero stress components are Tii, Ti2, T22, T33 where 17 Considering a static...

## F Experimental Measurement of Strain

90 Typically, the transducer to measure strains in a material is the strain gage. The most common type of strain gage used today for stress analysis is the bonded resistance strain gage shown in Figure 4.9. Figure 4.9 Bonded Resistance Strain Gage 91 These gages use a grid of fine wire or a metal foil grid encapsulated in a thin resin backing. The gage is glued to the carefully prepared test specimen by a thin layer of epoxy. The epoxy acts as the carrier matrix to transfer the strain in the...

## Hydrostatic and Deviatoric Strain

85 The lagrangian and Eulerian linear strain tensors can each be split into spherical and deviator tensor as was the case for the stresses. Hence, if we define then the components of the strain deviator E' are given by We note that E' measures the change in shape of an element, while the spherical or hydrostatic strain iel represents the volume change. es disusing lt C Jo h0 we t,ai t tOlst MatrixForm Tfirst . 0, 1, 0 gt We note that this vector is in the same direction as t e Cauchy stress...